# The Turning Point Test of Statistical Independence

### Description

Perform a test of statistical independence of a data series by comparing the number of turning points present in the series with the number of turning points expected to be present in an i.i.d. series.

### Usage

1 |

### Arguments

`x` |
a numeric vector or univariate time series. |

### Details

If the data is x[1], x[2], ..., x[n], then there is a turning point at the point i if either x[i-1]<x[i] and x[i]>x[i+1], or x[i-1]>x[i] and x[i]<x[i+1]. this function counts the number of turning points in the data, standardises it to have mean 0 and variance 1 and asymptotically tests it against a standard normal distribution. The test statistic is

T = (tp-mu)/sigma, where

tp is the number of turning points present in the series,

mu = 2*(n-2)/3,

sigma = sqrt((16*n-29)/90) and

n is the number of data points in the series.

The test is set up as follows:

*H0*: the data series is i.i.d. (not trending)

*H1*: the data series is not i.i.d. (trending)

### Value

A list with class "htest" containing the following components:

`statistic` |
the value of the test statistic. |

`p.value` |
the p-value of the test. |

`method` |
a character string indicating what type of test was performed. |

`data.name` |
a character string giving the name of the data. |

`n` |
the number of points in the data series. |

`mu` |
The expected number of turning points that would be seen in an i.i.d. series. |

`sigma` |
The standard deviation of the number of turning points that would be seen in an i.i.d. series. |

### Note

Missing values are not handled.

Points followed by a point having the exact same value are removed from the data series before computing the test statistic.

This test is useful for detecting cyclic/periodic trends in data series.

### Author(s)

Andrew Hart and Servet Mart<ed>nez

### References

Brockwell, Peter J., Davis, Richard A. (2002) *Introduction to Time Series and Forecasting*.
Springer Texts in Statistics, Springer-Verlag, New York.

Bienaym<e9>, Ir<e9>n<e9>e-Jules (1874).
Sur une question de probabilit<e9>s.
*Bull. Math. Soc. Fr.*
**2**, 153-154.

### See Also

`diffsign.test`

, `rank.test`

, `lb.test`

,
`markov.test`

, `diid.test`

### Examples

1 2 3 | ```
#Generate an IID standard normal sequence
n <- rnorm(1000)
turningpoint.test(n)
``` |

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